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Advanced Analytic Number Theory: L-Functions
Page17(38 of 313)

1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 17 / $(a)xO) 1^/i = n / ®v(av)Xv(av) 1d/jJV JG v JGV is a finite product, depending on x, each of whose factors is bounded by I / §v(av)Xv{av)~ldyiv\ / \$v(av)\dfj,v +00, JG„ JG„ and fJGv$v{avYv)xv(av)~ldnv G $v{a = 1 for almost all v. Hence the Fourier transform is a product function, and $(x) £ Li(G). With these preliminaries, and using the inversion formula applied to Gv and Gv, we obtain the following result. T H E O R E M 11. With notations as above, let $ : G — C be a continuous function with l G Li(G) and $ G L\{G). Then the inversion formula JG holds with the self dual Haar measure given by the product measure dx = Y[dxv. V The Schwartz-Bruhat space on a restricted direct product of locally compact groups {Gv} is the span of linear combinations of product functions V where each $v G S(GV) and $v — charH± for almost all v. T H E O R E M 12. The Fourier transform $ •- Hx) = / $(a)x(a)d//(a) induces an isomorphism from the Schwartz-Bruhat space S(G) onto the space S(G). In the main applications we deal with groups G that are self dual, i.e. G is isomorphic to G. In those situations the Fourier transformation induces an isomorphism on the Schwartz-Bruhat space S{G) which extends to the space of tempered distributions S'(G).